For the mapping \(F(\mathbf {q}^{k_m}, \mathbf {\eta }^{k_m})\)  defined in (REF ), let \(\partial F(\mathbf {q}^{k_m}, \mathbf {\eta }^{k_m})\)  be the generalized Jacobian of (REF ) in sense of Clarke (see [1]}). Then, as analyzed in [2]}, we have
\(F^{\prime }(\mathbf {q}^{k_m}, \mathbf {\eta }^{k_m}):= \begin{pmatrix}J^{\prime \prime }(\mathbf {q}^{k_m})+\beta M&I\\-c\Pi _{{\mathcal {A}}_{k_m}}& \Pi _{{\mathcal {I}}_{k_m}}\end{pmatrix} \in \partial F(\mathbf {q}^{k_m}, \mathbf {\eta }^{k_m}),\) 
